Noise Sensitivity and Noise Stability

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Noise Sensitivity and Noise Stability

• Gil Kalai
• Hebrew University of Jeusalem and Yale University
  
• NY-Chicago-TA-Barcelona 08-09

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• (We start with a one-slide summary of the
  lecture followed by a 4 slides very informal
  summary of its four main parts.)

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Plan of the talk

• 1) Planar percolation
• 2) Boolean functions, influences.
• Noise sensitivity The primal description
• Noise sensitivity - The Fourier description
• 3) Percolation noise sensitivity, the spectrum,
  and dynamic percolation
• 4) The Majority is Stabelest theorem, MAX CUT,
  and voting.

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Boolean functions and influences

• We consider a BOOLEAN FUNCTION
• f -1,1n ? -1,1
• f(x1 ,x2,...,xn)
• Boolean functions are of importance in
  combinatorics, probability theory, computer
  science and other areas.
• The influence of the kth variable xk on f is the
  probability that flipping the value of the kth
  variable will flip the value of f.

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Planar Percolation

• The infinite model we have an infinite lattice
  grid
• in the plane. Every edge (bond) is open with
• probability p. All these probabilities are
  statistically independent.
• Basic questions
• What is the probability of an infinite open
  cluster?
• What is the probability of an infinite open
  cluster containing the origin?
• Critical properties of percolation.

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Noise sensitivity

• Primal description - Functions (random
  variables) that are extremely sensitive to small
  random changes (which respect the overall
  underlying distribution.) Such functions cannot
  be measured by (even slightly) noisy
  measurements.
• Dual description Spectrum concentrated on
  large sets
• Examples Critical percolation, and many others
• Basic insight Noise sensitivity is common and
  forced in various general situations.
• The notions of noise stability and noise
  sensitivity were introduced by Benjamini, Kalai
  and Schramm. Closely related notions (black
  noise non-Fock models) were introduced by
  Tsirelson and Vershik.

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Two recent breakthroughs

• Garban, Pete and Schramm achieved wonderful
  understanding of noise sensitivity for critical
  percolation. Detailed understanding of the
  scaling limit for the spectral distribution, and
  of dynamic percolation followed.
• Mossel, ODonnell and Oleszkiewicz Proved that
  from all Boolean functions with diminishing
  influence the majority function is asymptotically
  most stable. This settled open problems regarding
  hardness of approximation for MAXCUT and the
  Condorcet Paradox.

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Critical Percolation
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• Part I
• Planar Percolation

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Critical Percolation problems and progress

• The critical probability
• Limit conjectures and Conformal invariance
• SLE and scaling limits
• Noise sensitivity and spectral description

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Kesten Critical probability 1/2

• Kestens Theorem (1980) The critical probability
  for percolation in the plane is ½.
• If the probability p for a bond to be open (or
  for a hexagon to be grey) is below ½ the
  probability for an infinite cluster is 0. If the
  probability for a bond to be open is gt ½ then the
  probability for an infinite cluster is 1.
• (Q And when p is precisely ½?)
• (A The probability for an infinite cluster is
  0)

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Limit conjectures

• Conjecture The probability for the crossing
  event for an n by m rectangular grid tends to a
  limit if the ratio m/n tends to a real number a,
  agt0, as n tends to infinity.
• (Sounds almost obvious, yet very difficult to
  prove)
• Note we have moved from infinite models to
  finite ones.

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Langlands, Pouliot, Saint-Aubin, Cardy, Aizenman
Conformal invariance conjectures

• Conjecture Crossing events in percolation are
  conformally invariant!!
• Sounds very surprising. (But there is no case of
  a planar percolation model where the limit
  conjectures are proven and conformal invariance
  is not.)

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Limits Conjectures and conformal Invariance
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Schramm SLE

• Oded Schramm defined a one parameter planar
  stochastic models SLE(?). Lawler, Schramm and
  Werner extensively studied the SLE processes,
  found relations to several planar processes, and
  computed various critical exponents. SLE(6)
  describes the scaling limit of percolation.

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SLE and PercolationGrey/white Interface
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Smirnov Conformal Invariance

• Smirnov proved that for the model of site
  percolation on the triangular grid, equivalently
• For the white/grey hexagonal model (simply
  HEX), the conformal invariance conjecture is
  correct!
• (An incredibly simple form of Cardys formulas
  in this case found by Carleson was of
  importance.)

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Putting things together

• Combining Smirnov results with the works of
  Lawler Schramm and Werner (and some earlier works
  of Kesten) all critical exponents for percolation
  predicted by physicists and quite a few more (and
  for quite a few other planar models) were
  computed. (rigorously)
• (For the model of bond percolation with square
  grid this is yet to be done.)

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• Part II
• Boolean Functions and Noise Sensitivity

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Boolean Functions

• We consider a BOOLEAN FUNCTION
• f -1,1n ? -1,1
• f(x1 ,x2,...,xn)
• It is convenient to regard -1,1n as a
  probability space with the uniform probability
  distribution.

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Influence

• We consider a BOOLEAN FUNCTION
• f -1,1n ? -1,1
• The influence of the kth variable xk on f,
  denoted by Ik(f) is the probability that flipping
  the value of the kth variable will flip the value
  of f.

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Examples

• 1) Dictatorship f(x1 ,x2,...,xn) x1
• Ik(f) 0 for kgt1 I1(f)1
• 2) Majority f(x1 ,x2,...,xn) 1
• iff
• x1 x2...xn gt 0
• Ik(f) behaves like n-1/2 for every k.

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Examples (cont.)

• 3) The crossing event for percolations
• For percolation, every hexagon corresponds to a
  variable. xi -1 if the hexagon is white and xi
  1 if it is grey. f1 if there is a left to right
  grey crossing.
• Ik(f) behaves like n-3/8 for every k but few.

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Noise Sensitivity The Primal description

• We consider a BOOLEAN FUNCTION
• f -1,1n ? -1,1
• f(x1 ,x2,...,xn)
• Given x1 ,x2,...,xn we define y1 ,y2,...,yn as
  follows
• xi yi with probability 1-t
• xi -yi with probability t

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Noise Sensitivity The Primal description (cont.)

• Let C(ft) be the correlation between
• f(x1 , x2,...,xn) and f(y1,y2,...,yn)
• A sequence of Boolean function (fn ) is
  (completely) noise-sensitive if for every tgt0,
  C(fn,t) tends to zero with n.

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• Part III
• Fourier Analysis, noise sensitivity and
  percolation

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Percolation is Noise sensitive

• Theorem BKS The crossing event for critical
  planar percolation model is noise- sensitive
• Basic argument 1) Fourier description of noise
  sensitivity 2) hypercontractivity
• This argument applies to very general cases.

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Percolation is Noise sensitive

• Imagine two separate pictures of n by n
  hexagonal models for percolation. A hexagon is
  grey with probability ½.
• If the grey and white hexagons are independent in
  the two pictures the probability for crossing in
  both is ¼.
• If for each hexagon the correlation between its
  colors in the two pictures is 0.99, still the
  probability for crossing in both pictures is very
  close to ¼ as n grows! If you put one drawing on
  top of the other you will hardly notice a
  difference!

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Fourier-Walsh expansion

• Given a Boolean function f -1,1n ? -1,1, we
  write f(x) as a sum of multilinear (square free)
  monomials.
• f(x) Sfˆ(S)W(S), where W(S) ?xs s ? S.
• f(S) is the Fourier-Walsh coefficient
  corresponding to S.
• Used by Kahn, Kalai and Linial (1988) to settle a
  conjecture by Ben-Or and Linial on influences.

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Noise sensitivity the dual Description

• The spectral distribution of f is a probability
  distribution assigning to a subset S the
  probability (f(S))2
• For a sequence of Boolean function
• fn -1,1n ? -1,1
• (fn) is noise sensitive if for every k the
  overall spectral probability for non empty sets
  of size at most k tends to 0 as n tends to
  infinity.

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Our theorem

• Thaorem (Benjamini, K. Schramm 1999) For a
  sequence of Boolean functions fn
• If H(f) the sum of squares of the influences
  tends to 0 then (fn ) is noise sensitive.
• (Easier a simple application of Beckners
  estimates.) If H(f) lt n-b for some bgto then most
  of the spectral distribution of f is above the
  log n level.

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The motivations

• This was an attempt towards limits and
  conformal invariance conjectures. (Second attempt
  with records for Oded and Itai.)
• Understanding the spectrum of percolation
  looked interesting One critical exponent
  (correlation length) has a simple description.
• (Late) Percolation on certain random planar
  graphs arises here naturally. (KPZ)

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An application Dynamic percolation

• Dynamic percolation was introduced and first
  studied by Häggström, Peres and Steif (1997).
  The model was introduced independently by Itai
  Benjamini. Häggström, Peres and Steif proved that
  above the critical probability we have infinite
  clusters at all times, and below the critical
  probability there are infinite clusters at no
  times.
• Schramm and Steif proved that for critical
  dynamic percolation on the HEX model there are
  exceptional times. The proof is based on their
  strong versions of noise sensitivity for planar
  percolation. They needed stronger results about
  noise sensitivity of percolation.

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Dynamic Percolation
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Fourier Description of Crossing events of
Percolation

• Benjamini, Kalai, and Schramm Most Fourier
  Coefficients are above log n
• Schramm and Steif Most Fourier coefficients are
  above nb (bgt0)
• Schramm and Smirnov Scaling limit for spectral
  distribution for Percolation exists ()
• Garban, Pete and Schramm Spectral distributions
  concentrated on sets of size n3/4(1o(1)). ()
• () proved only for models where Smirnovs
  result apply.
• The scaling limit for the spectral distribution
  of percolation is described by Cantor sets of
  dimension ¾.

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Garbon, Pete and Schramm

• Garban, Pete and Schramm Spectral distributions
  concentrated on sets of size n3/4(1o(1)). ()
• () proved only for models where Smirnovs
  result apply.
• The scaling limit for the spectral distribution
  of percolation is described by Cantor sets of
  dimension ¾.
• Towards a full understanding of the scaling limit
  for dynamic percolation.
• Much, much more

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An ingredient in the proof

• The first two moments of the spectral
  distribution coincide with those of the pivotal
  distribution. h(x) is the number of neighbors of
  x where f attains a different value.
• Sf2 (S)S SIk (f) S2-n h(x) (KKL)
• Sf2 (S)S2 S2-n h2 (x)

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Critical Percolation
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Other cases of noise sensitivity

• First Passage Percolation (Benjamini, Kalai,
  Schramm)
• A recursive example by Ben-Or and Linial (BKS)
• Eigenvalues of random Gaussian matrices
  (Essentially follows from the work of
  Tracy-Widom) Here, we leave the Boolean setting.
• Examples related to random walks (required
  replacing the discrete cube by trees) and more...

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• Part IV
• Majority is most stable Stabe

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Diversion Simulating and computing the spectrum
for percolation

• Can we sample according (approximately) to the
  spectral distribution of the crossing event of
  percolation?
• This is unknown and it might be hard on digital
  computers.
• But... it is known to be easy for... quantum
  computers. For every Boolean function where f is
  computable in polynomial time. (Quantum computers
  are hypothetical devices based on QM which allow
  superior computational power.)

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Majority is noise stable

• Sheppard Theorem (99)
• Suppose that there is a probability t for a
  mistake in counting each vote.
• The probability that the outcome of the election
  are reversed is arccos(1-t)/p

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Majority is noise stable (cont.)

• Sheppard Theorem (1899) Suppose that there is a
  probability t for a mistake in counting each
  vote.
• The probability that the outcome of the election
  are reversed is arccos(1-t)/p
• When t is small this behaves like t1/2

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Majority is noise stable (cont.)

• Suppose that there is a probability t for a
  mistake in counting each vote. The probability
  that the outcome of the election are reversed is
  arccos(1-t)/p
• When t is small this behaves like t1/2
• Is there a more stable voting rule? Sure!
  dictatorship

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Majority is stablest!

• Theorem Mossel, ODonnell and Oleszkiewicz
  (2005)
• Let (fn ) be a sequence of Boolean functions with
  diminishing maximum influence. I.e., lim max
  Ik(f) -gt 0
• Then the probability that the outcome of the
  election are reversed when for every vote there
  is a probability t it is flipped is at least
• (1-o(1)) arccos(1-t)/p

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Majority is stablest! Two applications

1. The probabilities of cyclic outcomes for voting
   rules with diminishing influences are minimized
   for the majority voting rule!
2. Improving the Goemans-Williamson 0.878567
   approximation algorithm is hard, unique-game-hard!

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The remarkable story of PCP and Hardness of
approximation is related to Fourier analysis of
Boolean functions. Khot Kindler Mossel and
Oddonell showed that majority is stablest implies
hardness of improving Goemans-Williamson MAX CUT
algorithm
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• This is where the talk officially ends Thank
  you!
• A little more on noise sensitivity follows.

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Noise sensitivity, and non-classical stochastic
processes black noise

• Closly related notions to noise sensitivity
  were studied by Tsirelson and Vershik . In their
  terminology noise sensitivity translates to
  non Fock processes, black noise, and
  non-classical stochastic processes. Their
  motivation is closer to mathematical quantum
  physics.

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Tsirelson and Vershik Non Fock spaces black
noise non classical stochastic processes (cont)

• The terminology is confusing but here is the
  dictionary
• Noise stable White noise classical stochastic
  process Fock model
• Noise sensitive Black noise non classical
  stochastic process non-Fock model.
• Tsirelson and Vershik pointed out a connection
  between noise sensitivity and non-linearity.
  (Well within the realm of QM.)

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Questions about HEP models

• Are current HEP models noise stable?
• Or perhaps there is some internal inconsistency
  about their noise stability
• The naive idea is this Hep models describe a
  (quantum) stochastic state. Is this state
  necessarily noise stable?
• (Less naively, according to Tsirelson) Noise
  sensitivity means that the very idea of the
  field operator at a point' (on the level of
  operator-valued Schwartz distributions (or
  something like that)) will fail.

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Questions about HEP models

• Tsireslon constructed a toy non-Fock model in
  hep-th/9912031
• My thoughts on the matter can be found in
• hep-th/0703092